William L. Fink’s research while affiliated with University of Michigan and other places

This chapter aims to describe a variety of patterns that can be found in comparative studies of ontogeny. To that end, it focuses on the patterns amenable to discovery by comparative studies of ontogenetic allometry. Studies of allometry are sometimes viewed as a poor substitute for studies of heterochrony, but allometry is not just something we study when we have no information about age. Rather, comparative studies of allometry allow for a richer formalism than is feasible in studies of heterochrony because the formalisms for heterochrony were designed for cases of parallelism. They cannot be applied more generally without sacrificing a multivariate approach to the evolution of ontogeny. Comparative analyses of allometry not only rely on a richer formalism; they also analyze a phenomenon that is interesting in its own right. Allometry is no less interesting than heterochrony, an argument presented in the chapter. After motivating the study of ontogenetic allometry, the chapter introduces the formalism for analyzing it and discusses the meaning (both formal and biological) of the coefficients obtained by that formalism. It next discusses methods for discerning patterns in the relationship between ontogeny and phylogeny, focusing on the significance of those patterns for our understanding of the evolution of development. The first part of the chapter focuses on traditional morphometric data because most comparative studies of allometry have relied on them. It then briefly reviews the geometric analysis of ontogenetic allometry and revisits the patterns introduced in the context of traditional allometry, describing how these would appear in studies of geometric shape data.

Systematists use morphometrics to answer three types of questions. The first, “taxonomic,” asks whether populations are drawn from multiple species, and, if so, by what variable(s) they are most effectively discriminated. The second, “phylogenetic,” asks about phylogenetic relationships among taxa. Although they cannot be used to construct cladograms, morphometric analyses might nonetheless be useful for finding informative characters. The third, “evolutionary,” asks about the evolutionary history of the feature of interest—which, for our purposes, is shape. These are all interrelated issues, but there are important distinctions that bear on choosing the appropriate analytic method. Most importantly, taxonomic discriminators are often not equivalent to phylogenetically informative characters, so finding discriminators is not equivalent to finding characters. Also, characters usually comprise a subset of features that evolve, so tracing characters on a cladogram does not fully reconstruct the evolution of shape. Unfortunately, of the three types of questions, only those relating to taxonomic discrimination are so straightforward that they require nothing more than standard morphometric tools. This does not mean taxonomic discrimination is easy; on the contrary, it can be very difficult.

This chapter discusses two methods for describing the diversity of shapes in a sample: principal components analysis (PCA) and canonical variates analysis (CVA). The discussion of these methods draws heavily on expositions presented by Morrison (1967), Chatfield and Collins (1980), and Campbell and Atchley (1981). Both methods are used to simplify descriptions rather than to test hypotheses. PCA is a tool for simplifying descriptions of variation among individuals, whereas CVA is used for simplifying descriptions of differences between groups. Both analyses produce new sets of variables that are linear combinations of the original variables. They also produce scores for individuals on those variables, and these can be plotted and used to inspect patterns visually. Because the scores order specimens along the new variables, the methods are called “ordination methods.” The most important difference between PCA and CVA is that PCA constructs variables that can be used to examine variation among individuals within a sample, whereas CVA constructs variables to describe the relative positions of groups (or subsets of individuals) in the sample. PCA and CVA both serve a similar purpose, and the mathematical transformations performed in the two analyses are similar. The chapter describes PCA first because it is somewhat simpler, and because it provides a foundation for understanding the transformations performed in CVA. It begins the description of PCA with some simple graphical examples, and then presents a more formal exposition of the mathematical mechanics of PCA. This is followed by a presentation of an analysis of a real biological data set.

Landmarks are discrete anatomical loci that can be recognized as the same loci in all specimens in the study. As landmarks play a fundamental role in geometric morphometrics, it is important to understand their function in a shape analysis. It is equally important to understand which functions they do not serve, as that understanding also influences the selection of landmarks. Criteria for selecting landmarks differ from those applied to choosing traditional morphometric variables, so some rethinking may be required. The chapter begins with a summary of some of the basic differences between conventional and landmark-based studies that bear on landmark selection and on how we may need to change the way we think about selecting variables. Next is a review of the criteria for choosing landmarks in light of both biological and mathematical considerations, focusing on general criteria and principles. This is followed by three concrete examples, each explaining why particular landmarks were chosen. The chapter concludes with a practical guide to collecting landmark data.

This chapter presents a brief discussion of some of the basic statistical concepts that are needed to understand statistical methods in general (such as confidence intervals and hypothesis testing), as well as the more specialized concepts that are needed to understand computer-based statistical methods. Four classes of these methods are presented, including the bootstrap, jackknife, and permutation tests, and Monte Carlo simulations. To illustrate these methods, the chapter focuses on a few univariate statistical tests. The extension to multivariate statistics is not difficult, but it seems useful to focus on univariate statistics to develop an intuitive understanding of how computer-based methods work.

Many structures of interest to biologists are three-dimensional, or have few landmarks, or both. The skull of a marmot, like that of most mammals, is an example of “both.” The marmot skull is strongly curved anteroposteriorly and mediolaterally, making it highly three-dimensional (features on the same bone may be as far apart in the dorsoventral dimension, as they are in the mediolateral or anteroposterior dimensions). In addition, the skull is composed of a small number of relatively large bony plates, so points that can be used as landmarks are sparsely distributed, occurring primarily at locations where at least three bones meet. The first part of this chapter examines methods that have been devised to analyze three-dimensional configurations of landmarks, and the second examines methods that have been devised to analyze curves and surfaces that lack landmarks. The chapter discusses the general problems and the advantages and disadvantages of particular approaches.

Disparity and variation are closely allied concepts—both refer to the general idea of “variety.” Disparity usually signifies the variety of a group of species and is the outcome of evolutionary processes; variation, on the other hand, refers to the variety of individuals within a single (homogeneous) population and is the raw material necessary for evolution. In light of the theoretical distinction between the two concepts, it may seem difficult to cover both in a single chapter. However, the distinction between the concepts lies in the processes that produce them and the theories that predict them. The metric (or formula) for measuring disparity among species is the same as that used to measure variation within a species. Since the same metric is used to measure both, both of them are covered in the same chapter. Even so, to avoid confounding concepts that have little in common aside from a metric, the chapter begins by reviewing their biological meanings, then turns to the issue of measurement.

This chapter covers methods for testing hypotheses about samples that vary along a continuously valued factor—a factor measured on an infinitely divisible scale. Size is an example of such a continuously valued factor because there is always a size between any two others; similarly, latitude is continuously valued because there is a latitude between any two others. When we hypothesize that a continuously valued factor affects the shape, we use regression to test the hypothesis. Additionally, when we want to control for the effects of such a factor so that we can distinguish between groups defined by a categorical variable, we use regression to control for those effects. Finally, we would use regression when our hypotheses concern the particular nature of an effect, i.e., the direction of the shape variable covarying with the factor of interest. For example, if our hypothesis is that two species follow a common ontogeny of shape, we use regression to describe each ontogeny, then we compare the two vectors, asking if they point in the same direction. The chief aim of regression is to explain the variation in one variable (shape, in our case) by another. For example, we might suspect that several factors account for the variation in our data, including age or size; geographic variables such as latitude, longitude or temperature; ecological variables such as the size of predators and the density of the canopy; or even clinically important characteristics such as health status.

This chapter begins with a brief review of groups and grouping variables. It then presents the simplest case, the test for a difference in one trait between two groups, and the methods that would be used in such cases. It follows this with a series of more complex analyses and the more generalized methods that would be applied to them. The final section presents instructions for performing the analyses discussed in the chapter. A group is a set of individuals (a class) defined as sharing a state of a discontinuous trait. In mammals and birds, “sex” is an example of a discontinuous trait that has two classes—“male” and “female.” An individual is either one or the other as a consequence of having one set of chromosomes or the other. Such traits may be called grouping variables, qualitative traits or categorical variables. All these names refer to the fact that the states of the trait do not have intrinsic numerical values or an inherent order, but they can nonetheless be used to sort individuals into groups or categories.

This chapter presents a method for obtaining shape variables that is both simple and visually informative. Called “the two-point registration,” this method produces a set of shape coordinates, sometimes called “Bookstein shape coordinates,” that can be used both for graphical displays and formal statistical tests. Bookstein shape coordinates (BC) provide a useful introduction to shape analysis because they are intuitively accessible, their formula is relatively straightforward, and understanding them does not require a general understanding of morphometric theory. To introduce the two-point registration, the chapter first reviews the meaning of shape because this meaning is crucial to the formula. It then focuses on the simplest possible application of the method, the analysis of shapes with only three landmarks (triangles). It also discusses how information about the size can be restored (because it is removed in the course of the two-point registration). Once we have shape coordinates and a measure of size, we can then test the hypothesis that two samples of shapes differ statistically or that shape change is correlated with size change. These statistical tests are done directly on the coordinates of landmarks—should a statistically significant difference (or covariance) be found, one can then depict it and describe the variable that differs or changes. The chapter also discusses the description of shape variables and the biological interpretation of them, because, to a large extent, it is the descriptive power of geometric morphometrics that makes these methods so useful.